Chirp signals are used in a variety of applications, from radar and oil well logging to electronic and mechanical systems analysis. For expository convenience, the following discussion focuses on one particular application--the use of chirp signals by dynamic signal analyzers to analyze electronic and mechanical systems, although it will be recognized that the invention is not so limited.
Basically, a chirp signal is a sinusoid that sweeps rapidly from one frequency to another. In dynamic signal analyzers, such signals are used to stimulate a mechanical or electronic system under study. The system's response to the chirp can be analyzed to predict the system's response to a multitude of other conditions for which individual testing is impractical. In this particular application, the chirp is typically repeated periodically, with one chirp beginning immediately after the previous chirp is completed.
One desirable feature of chirp signals in this test application is their small crest factor. Crest factor is the ratio of a waveform's peak instantaneous values to its rms value. A one volt rms chirp would typically have a peak voltage of 1.414 volts, so the crest factor is 1.414. (In contrast, another commonly used test waveform, white noise, has a crest factor of three.) In analyzing a system, it is desirable to use as large an input signal as possible in order that the system's response to the signal not be masked by noise in the system or in the instrumentation. However, if the stimulus signal exceeds a certain peak value, the system will distort, corrupting the resulting measurements. Thus, the peak-to-rms ratio for a given waveform determines how much signal power can be provided to the system without it distorting. All else being equal, a chirp will provide more stimulus energy to a system than any other wideband signal type.
More important than the crest factor in most analyses is the requirement that the spectral distribution of energy in the test signal be constant throughout its frequency range. If it is not, then subsequent analysis cannot determine whether a system's particularly strong response at one frequency was due to a resonance at that frequency or whether the stimulus signal simply had a local maximum at that frequency component, prompting an exaggerated response. Another way of stating the requirement is that the test signal should have a perfectly flat response in the frequency domain. While simple to conceptualize in the case of a chirp signal, it is not simple to realize.
The problem with realizing a flat response in the frequency domain is the finite duration of the chirp. An infinite chirp, characterized in the frequency domain by the formula e.sup.jkw.spsp.2, has a perfectly flat response, as does its counterpart in the time domain, e.sup.jkt.spsp.2. However, when the chirp is truncated, its frequency domain representation takes on a sin(x)/x ripple--sin(x)/x being the Fourier transform of the finite truncation operator. (Actually the frequency domain response is somewhat further disturbed because the chirp isn't just truncated to a finite section, but then that section is repeated. However, the dominant effect is the sin(x)/x ripple.) In a typical truncated chirp, different frequency components may differ in magnitude by 6 decibels. This large ripple has generally been considered unavoidable and has simply been tolerated in the prior art, with attendant anomalous results in certain measurements.
According to the present invention, a chirp signal is provided that is characterized by a flat response in the frequency domain. To achieve this response, the signal is made non-flat in the time domain. That is, its amplitude is varied as its frequency is swept. The particular perturbations introduced into the chirp's waveform are typically selected to minimize the waveform's crest factor.
In one method according to the present invention, a chirp signal is produced by generating a sequence of samples according to the formula e.sup.jkw.spsp.2, performing a finite inverse Fourier transform on the sequence to find its counterpart in the time domain, storing digital data corresponding to the transformed sequence in a memory, and then reading the digital data from memory. The resulting digital data stream can be converted directly into analog form to produce a chirp at its baseband frequency, or the digital data stream can be mixed with a digital local oscillator and subsequently converted to analog form to produce a chirp at another, typically higher frequency.
The foregoing and additional features and advantages of the present invention will be more readily apparent from the following detailed description, which proceeds with reference to the accompanying drawings.